Geodesic
The Kramers--Wannier Symmetry and $S$-Duality in the Two-Dimensional $g\Phi ^4$ Theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 2, pp. 206-215

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We show that the exact beta function of the two-dimensional $g\Phi ^4$ theory possesses two dual symmetries. These are the Kramers–Wannier symmetry $d(g)$ and the strong-weak-coupling symmetry, or the $S$-duality $f(g)$, connecting the strong- and weak-coupling domains lying above and below the fixed point $g^*$. We obtain explicit representations for the functions $d(g)$ and $f(g)$. The $S$-duality transformation $f(g)$ allows using the high-temperature expansions to approximate the contributions of the higher-order Feynman diagrams. From the mathematical standpoint, the proposed scheme is highly unstable. Nevertheless, the approximate values of the renormalized coupling constant $g^*$ obtained from the duality equations agree well with the available numerical results. 
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     author = {B. N. Shalaev},
     title = {The {Kramers--Wannier} {Symmetry} and $S${-Duality} in the {Two-Dimensional} $g\Phi ^4$ {Theory}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {206--215},
     publisher = {mathdoc},
     volume = {131},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_131_2_a3/}
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B. N. Shalaev. The Kramers--Wannier Symmetry and $S$-Duality in the Two-Dimensional $g\Phi ^4$ Theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 2, pp. 206-215. http://geodesic.mathdoc.fr/item/TMF_2002_131_2_a3/